Energy eigenvalues (English)

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3-1: Energy Eigenvalues and Eigenstates

We learned on the page,

4-3: The Bohr Model of Atoms,

in the previous Seminar

that the atomic structure

can beautifully be described

by Bohr's old quantum theory,

in which it was hypothesized

that only

the discrete

(step-like) values of

energy are allowed

for the atomic stationary states

and

these states

are determined

by the quantum condition.

We learned that this

old quantum theory

can well explain

the structure

of hydrogen atom.

Then, we have

a question whether

Bohr's old quantum theory

can be derived

from quantum mechanics.

Does the Schroedinger equation

bring out

the stationary states

with discrete energies?

[Energy Eigenvalues of Bound States]

We learned on the page, 1-6: Energy Eigenvalues, Eigenstates,

that, if we treat a harmonic

oscillator (spring vibration)

with quantum mechanics,

we could obtain

the eigenstates

with the discrete

energy eigenvalues.

Such discrete

or step-like eigenvalues

are not only

in a harmonic oscillator.

These are a common property

of the states called

"bound states"

in quantum mechanics.

This property is never

seen in the classical theory.

What is its origin?

Let us briefly

explain this below.

Suppose a potential

well shown in the above

Fig. (A).

Consider the motion

of a particle of mass m

satisfying

the Schroedinger equation

Here we consider

a state in which the particle is

bound in the

square-well potential.

As seen in the figure,

the energy E

of the state must be

Let us see the feature

of the wave function

in the region of

a long distance

where the potential

vanishes,

i.e., V (x )

= 0.

In this region,

the Schroedinger equation

(1)

becomes

Accordingly, the

general solution

of the wave function

in the region of

V (x )

= 0

is written

where

and B

are arbitrary constants.

Since the square

of the absolute value

of the wave function

denotes the probability density

that the particle

will be found,

the wave function

must not diverge at any point

in the space.

Accordingly, in the wave function (4),

the constant A

must be 0 at the distant right

and the constant B

must be 0 at the distant left ,

i.e.,

the wave function must be

In order that

the boundary condition

that the wave functions

do not diverge

is satisfied,

the energy E

cannot be arbitrary

but must be of some

specific values.

Thus the allowed energies

are not continuous but discrete.

Such specific discrete

(step-like) energies

are called

energy eigenvalues,

which was derived

from the German word eigen

meaning "characteristic"

or "unique".

We call this type of energies

discrete energy eigenvalues

or say that the energy

is quantized.

In addition,

the state corresponding

to each energy eigenvalue

is called the eigenstate.

The wave functions

of these eigenstates

vanishes at the long distance ().

Since almost all part

of the wave functions

are bound (or localized)

in the potential region,

these eigenstates

are called bound states.

We can, therefore, say that

the energy eigenvalues of bound states are always discrete.

The energy eigenvalues

and the corresponding wave functions,

which are sometimes

called the eigenfunctions,

are obtained by solving

the Schroedinger equation

(1)

under the boundary condition

(5).

Two examples are shown

in the following figures,

Fig. (B)

and

Fig. (C),

the former of which

is for the case of

a square-well potential and

the latter for a harmonic oscillator.

For the details

of the method to solve

these problems,

refer to an appropriate textbook

of quantum mechanics.

The eigenstates in the case of a square-well potential

The energy eigenvalues are

represented by the heights of

the horizontal levels

which are obtained

by solving the

Schroedinger equation (1).

The corresponding

wave functions

(thick solid curves)

are also shown.

The yellow-colored

part denotes the potential wall.

The number of the

possible bound states

depends on the value of .

The eigenstates of a harmonic oscillator

The energy eigenvalues are

represented by the heights

of the horizontal levels

which are obtained

by solving the

Schroedinger equation (1).

The corresponding

wave functions

(thick solid curves)

are also shown.

The yellow-colored

part denotes the potential wall.

The energy eigenvalues

measured from

the ground-state (0)

are integral multiples of

This is a distinctive feature

of the harmonic oscillator.

[The Reason of the Existence of Discrete Eigenvalues]

We should pay attention

to the reason

why such discrete (step-like)

eigenvalues appear.

This is because

a particle motion accompanies

the wave function.

In other words,

the eigenstates

with discrete energy eigenvalues

occur because of the

particle-wave duality.

This is extremely surprising and

never happens

in the classical theory.

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